The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations

ABSTRACT

We consider a natural generalization of the Nagell–Ljunggren equation to the case where the qth power of an integer y, for q ⩾ 2, has a base-b representation that consists of a length-ℓ block of digits repeated n times, where n ⩾ 2. Assuming the abc conjecture of Masser and Oesterlé, we completely characterize those triples (q, n, ℓ) for which there are infinitely many solutions b. In all cases predicted by the abc conjecture, we are able (without any assumptions) to prove there are indeed infinitely many solutions.

KEYWORDS:

• Nagell–Ljunggren problem
• Diophantine equation
• base-b representation
• repetition
• power
• abc-conjecture

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 11D61
• Secondary 11A63
• 11Y50
• 11D41

Acknowledgments

We thank the referee for several useful suggestions.

Additional information

Funding

Robert J. Lemke Oliver is supported in part by NSF Grant DMS-1601398. Jeffrey Shallit is supported in part by a grant from NSERC.